Reliable quantum master equation of the Unruh-DeWitt detector

TL;DR

This work addresses the long-standing question of when the quantum Markovian master equation accurately describes the open dynamics of Unruh-DeWitt detectors interacting with quantum fields. By adopting a relaxed late-time (van Hove) limit and expanding the evolution in the weak-coupling parameter $g$, the authors derive explicit memory-error estimates of order $\mathcal{O}(g^4)$ and establish concrete reliability conditions for the Markov approximation and the rotating wave approximation (RWA) in terms of the bath-correlation-derived quantities $\Delta_{\Omega}$ and $C_{\Omega}$. They show that CP-valid GKSL dynamics require RWA, while Markov validity imposes bounds $g^2|d\Delta_{\Omega}/d\omega|\ll1$ and $g^2|dC_{\Omega}/d\omega|\ll1$, with the timescales $t_{Markov}$ and $t_{RWA}$ set by these derivatives. The framework is illustrated with an accelerating UDW detector (Unruh effect), analyzing both massless and massive field cases: in the massless scenario, $C_{\Omega}=(\omega/4\pi)\coth(\pi\omega/a)$ and $\Delta_{\Omega}\approx (\omega/2\pi)\log(e^{\gamma}\omega\varepsilon)$, yielding a Gibbsian asymptotic state at the Unruh temperature $T=a/(2\pi)$; reliability requires weak enough coupling and appropriate detector size $\varepsilon$. Overall, the paper provides practical, $g$-dependent criteria that quantify when QMME predictions are trustworthy for UDW detectors across different spacetimes and field types, highlighting limitations and guiding experimental parameter choices.

Abstract

In this paper, we present a method for estimating the validity range of the quantum Markovian master equation as applied to the Unruh-DeWitt (UDW) detector within a broader context, particularly without necessitating an exact solution for the detector's evolution. We propose a relaxed van Hove limit (i.e., late-time limit) and offer a perturbative estimate of the error order resulting from the standard derivation procedure of open quantum dynamics. Our primary findings include reliability criteria for the Markov approximation and conditions for the applicability of the rotating wave approximation. Nevertheless, the specific forms of these validity conditions rely on the details of the detector-field system, such as the spacetime background, the trajectory of the detector, and the type of quantum field being analyzed. Finally, we illustrate our results by reexamining the open dynamics of an accelerating UDW detector undergoing the Unruh effect, where the validity conditions narrow the parameter space to ensure the solution's reliability regarding the quantum Markovian master equation.

Figures (3)

• Figure 1: Schematic diagram of the evolution of $\rho_d(t)$ governed by Eq.(\ref{['eq2.7']}) under weak coupling. By taking the weak coupling limit $g \rightarrow 0$, the evolution of the detector is only salient once $t$ approaches $\mathcal{O}\left(1 / g^2\right)$, which suggests a genuine limit $g^2 t \sim \mathcal{O}(1)$ while simultaneously taking $g \rightarrow 0$ and $t \rightarrow \infty$, should be taken to guarantee the convergence of Eq.(\ref{['eq2.7']}) sec1-13sec1-14.
• Figure 2: The dependency of the functions ${d C_{\Omega}}/{d \omega}$ (red dashed line) and ${C_{\Omega}}/{\omega}$ (blue solid line) on $\omega / a$ in the massless case. Both functions approach $1 / 4 \pi$ at the limit $\omega \gg a$. In the region of $\omega \ll a$, ${d C_{\Omega}}/{d \omega}$ approaches zero, while ${C_{\Omega}}/{\omega}$ diverges. This contradiction can be reconciled with a small coupling constant to uphold the Markov approximation and RWA simultaneously.
• Figure 3: (a) For different values of $m / a$, ${d C_{\Omega}}/{d \omega}$ varies as a function of $\omega / a$. All the curves approach zero when $\omega \ll a$, and asymptotically converge to $1/4 \pi$ for sufficiently large $\omega \gg a$. (b) For different values of $m / a$, ${C_{\Omega}}/{\omega}$ varies as a function of $\omega / a$. All the curves diverge at $\omega / a \rightarrow 0$, indicating that without sufficient suppression by a small coupling constant, the RWA may fail in this region. (c) For different choices of $\omega / a$, ${d C_{\Omega}}/{d \omega}$ varies as a function of $m / a$. All the curves approach zero for $m/a \gg 1$, indicating the Markov approximation is reliable for heavy quantum field background. (d) For different choices of $\omega / a$, ${C_{\Omega}}/{\omega}$ varies as a function of $m / a$. All the curves approach zero for $m/a \gg 1$, indicating the RWA is applicable for heavy quantum field background.